Cox Regression Review

Capstone 6940

Hazardous Conditions

Intro to Cox Regression

  • Cox regression is a popular regression modeling method used to predict survival rates given certain covariates.

  • It assumes that the effects of different variables on the outcome, survival, are constant over time.

    • “Survival” can refer to the development of a symptom, time to relapse after remission, or as a time to death [1].

    • Cox regression model is based on the hazard function h_x(t) with covariates at time t given by:

      h_x(t)=h_0(t)\exp(\beta_1x_1 +\beta_2x_2 + \dots + \beta_p x_n) h_x(t) = hazard \ function\ h_0(t) = Baseline \ hazard \ function\

      \beta_1x_1 +\beta_2x_2 + \dots + \beta_p x_n = linear\ combination\ of \ covariates\ and\ their\ coefficients

Hazard Ratios and Proportional Hazards

  • The hazard function is the probability that an individual will experience an event (death) within a certain time interval [1].

  • If the risk factor is binary, the result can be interpreted as the estimated increase in HR in patients with the risk factor vs. those without [2].

  • If the risk factor is continuous, the HR is interpreted as an increase/decrease in the hazard rate of death due to a 1 unit increase in the variable [2].

  • The assumption of a constant relationship between dependent and explanatory variables is called proportional hazards.

Testing Proportional Hazards

  • Graphical strategies to assess proportionality assumption

    • Kaplan-Meier Survival Distribution S(t): Plot S(t) as a function of survival time for each level covariate.

    • plot the function log(-log(S(t))) as a function of the log survival time (log represent natural log).

    • Schoenfeld Residuals

Time-Varying Coefficients

  • Failing to meet the assumption of proportional hazards means that the effects between dependent and explanatory variables are not constant over time.

  • Time-varying covariates (coefficients) are used when a covariate changes over time during the follow-up period [3].

    • Example: The effect of the size of a patient’s tumor on their chances of survival.

  • Internal time-varying coefficients are affected by survival status and include values that are generated by the subject [3].

    • A patient’s blood pressure levels during a study on cardiovascular events.

  • External time-varying coefficients are pre-determined and external to the subject under study [3].

    • Pollen levels during a study on patients with asthma.

References

[1]
S. J. Walters, “Analyzing time to event outcomes with a cox regression model,” Wiley Interdiscip. Rev. Comput. Stat., vol. 4, no. 3, pp. 310–315, May 2012.
[2]
S. Abd ElHafeez, G. D’Arrigo, D. Leonardis, M. Fusaro, G. Tripepi, and S. Roumeliotis, “Methods to analyze time-to-event data: The cox regression analysis,” Oxid. Med. Cell. Longev., vol. 2021, no. 1, p. 1302811, Nov. 2021.
[3]
Z. Zhang, J. Reinikainen, K. A. Adeleke, M. E. Pieterse, and C. G. M. Groothuis-Oudshoorn, “Time-varying covariates and coefficients in cox regression models,” Ann. Transl. Med., vol. 6, no. 7, pp. 121–121, Apr. 2018.